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%\noindent{\bigrm Math 1720 \hfill  Project 3}
%\medskip
\centerline{\bigrm Fibonacci Numbers.}
\bigskip

You may have heard about the Fibonacci sequence of numbers.  It starts
with the numbers
$$1,1,2,3,5,8,13,21,34,55,89,\cdots$$
\noindent
Even if you
haven't seen them before you may recognize the pattern.  To get each of
the numbers after the second one you simply add the two previous
numbers.  This gives an easy way to write out the Fibonacci sequence as
far as you wish.  Suppose that you only wanted to know the
$100^{\rm th}$ term of the sequence and you did not necessarily
want to know all the
numbers that came before it.  Is there a formula which would give you
this number directly?  In this project you will discover the formula.
Follow the steps below:
\medskip

\item{a.}Let $a_1,a_2,a_3,\cdots $ represent the Fibonacci sequence.
         Consider the power series $f(x)= a_1x+a_2x^2+a_3x^3+\cdots $.
         Show that the radius of convergence of this power series is at
         least $1\over 2$.
\item{b.}Find a polynomial $p(x)$ such that $p(x)f(x)=x$.
\item{c.}Find the power series expansion for $1\over ax+b$ and determine
         its radius of convergence.
\item{d.}Find the partial fraction decomposition for $x\over p(x)$
         and use it
         to determine the coefficients for $f(x)$.
\item{e.}What is the radius of convergence for $f(x)$?
\item{f.}Use your result from d) to determine a formula for $a_k$ for
         each value of $k$.  Use a calculator or computer to verify that
         the formula you derived is correct for small values of $k$.
         Then determine the value of $a_{100}$ and a few other
         values of $a_k$ for $k$ large.

\bigskip

You will be expected to justify every statement you make.  For example,
it is not good enough to just find a polynomial in part b).  You are
required to find the polynomial and then convince the skeptical reader
that $p(x)f(x)=x$.  As always, if you get stuck on a point then see your
instructor for a hint.
\bye
